Anomalies in Quantum Field Theory and Cohomologies of Configuration Spaces

TL;DR

This paper explores the mathematical structures underlying anomalies in quantum field theory by linking renormalization in configuration spaces to cohomology theories, offering new insights into the computation of renormalization group actions.

Contribution

It introduces a cohomological framework for analyzing renormalization anomalies, connecting them to de Rham cohomologies of configuration spaces and deriving differential equations for renormalization cocycles.

Findings

Cohomology spaces reduce to de Rham cohomologies of configuration spaces

Derived differential equations determine renormalization cocycles

Provides a new approach for computing renormalization group actions

Abstract

In this paper we study systematically the Euclidean renormalization in configuration spaces. We investigate also the deviation from commutativity of the renormalization and the action of all linear partial differential operators. This deviation is the source of the anomalies in quantum field theory, including the renormalization group action. It also determines a Hochschild 1-cocycle and the renormalization ambiguity corresponds to a nonlinear subset in the cohomology class of this renormalization cocycle. We show that the related cohomology spaces can be reduced to de Rham cohomologies of the so called "(ordered) configuration spaces". We find cohomological differential equations that determine the renormalization cocycles up to the renormalization freedom. This analysis is a first step towards a new approach for computing renormalization group actions. It can be also naturally extended to manifolds as well as to the case of causal perturbation theory. (This paper is based on previous preprint arXiv:0712.2194, but it has been entirely re-written and various new results are included.)